1. Field of the Invention
This invention relates to a circuit for measuring the conductivity of a solution disposed between two electrodes. More particularly, this invention relates to circuits for accurately measuring the conductivities of solutions varying over several orders of magnitude, wherein inherent sources of parallel and series capacitance are accurately compensated for.
2. Description of the Prior Art
There are many applications wherein it is desired to measure the electric conductivity of a solution. The conductivity of a solution is a measure of the dissolved ionic content of the solution. In low conductivity solutions, ranging from ultra-pure water used in semi-conductor and pharmaceutical manufacturing to power plant cooling water and potable water, the conductivity is measured as an indication of ionic impurities. In higher conductivity solutions, such as process chemicals and the like, conductivity is often measured to monitor and control the addition of ionic additives. In each of these applications there is a distinct need for apparatus and methods to accurately measure the conductivity of water. Measurements of the conductivity of solutions are relevant in many other industries and applications.
The conductivity of a solution can be determined by measuring its electrical resistance. Due to the nature of ionic solutions, however, measuring this resistance with dc current will cause ion migration that can substantially affect the measurement. For this reason, ac current is generally employed, at a voltage low enough and a frequency high enough to not affect the solution during the measurement.
The volume resistivity or simply the `resistivity` of a solution is defined as the resistance of one cubic centimeter of the solution at a specific temperature. The units of resistivity are ohm-cm (.OMEGA.-cm), kilohm-cm (K.OMEGA.-cm), or megohm-cm (M.OMEGA.-cm). Resistivity may be measured directly by applying an alternating current I.sub.c through the cell and measuring the resulting voltage drop V.sub.c across the electrodes. The resistivity .rho. is then: EQU .rho.=V.sub.c /KI.sub.c
where:
.rho. is the solution resistivity, in .OMEGA.-cm PA1 I.sub.c is the current applied through the cell, in amperes PA1 V.sub.c is the voltage measured across the cell, in volts PA1 K is the cell constant. PA1 V.sub.c is the voltage applied across the cell, in volts PA1 I.sub.c is the measured current through the cell, in amperes PA1 K is the cell constant.
The volume conductivity of a solution, also known as `specific conductance`, is defined as the inverse of the resistance of one cubic centimeter of the solution at a specific temperature. The units of specific conductivity are mho/cm (also known as Seimens) and .mu.mho/cm (.mu.Seimens, or .mu.S). Conductivity may be measured directly by applying an alternating voltage source V.sub.c across the cell and measuring the resulting current I.sub.c thereon. The specific conductance .sigma. is then: EQU .sigma.=KI.sub.c /V.sub.c
where: .sigma. is the specific conductance, in mho/cm
In either case, the basic parameter measured is the actual resistance of the solution, R.sub.X =V.sub.c /I.sub.c. Accurate measurement of R.sub.X is complicated by the presence of a parallel capacitance C.sub.P across the cell and a series capacitance C.sub.X formed at the solution-cell interfaces.
FIG. 1 depicts an approximate equivalent circuit of the solution-electrode interface. The solution resistance of interest is depicted as R.sub.X. Each electrode-solution interface forms an imperfect `double-layer` capacitance C.sub.d with an effective series resistance R.sub.c and an effective leakage resistance R.sub.d. Additionally, a capacitance C.sub.s is formed by the surface area of the electrodes 14 and 16 separated by the solution, acting as a dielectric.
FIG. 2 depicts a simplified equivalent circuit of the cell. The circuit parameter of analytical interest is R.sub.X, the resistance of the solution, primarily responsive to the ions in the solution. C.sub.P is the total effective parallel capacitance existing between the two electrodes, including any cable capacitance. The value of C.sub.P is substantially proportional to the area of the electrodes and inversely proportional to the separation of the electrodes. C.sub.P typically varies from less than 100 pf to over 1000 pf depending on cell geometry. C.sub.X is the total capacitance in series with R.sub.X, approximately equal to C.sub.d /2, and is again a function of cell geometry, generally increasing with increasing electrode surface area. C.sub.X typically ranges from 1 to 10 .rho.f.
FIG. 3 is a graph of the actual resistance R.sub.X (actual) of the solution in the cell versus the resistance observed R.sub.X (meas), that is, if no compensation is made for the contributions of C.sub.X and C.sub.P. As R.sub.X (actual) gets lower, i.e., in a solution containing larger numbers of ions, the impedance of C.sub.X becomes a larger proportion of R.sub.X (actual) and introduces an error indicated on FIG. 3 as "C.sub.X error". As R.sub.X (actual) gets higher, in less-ionic solutions, the parallel impedance of C.sub.P progressively reduces the measured impedance and introduces the error indicated as "C.sub.P error".
FIG. 4 is a graph of the conductivity .sigma..sub.x =1/R.sub.X of the sample in the cell, indicating the deviation of the conductivity .sigma..sub.X (actual) from the value .sigma..sub.X (meas) of the conductivity as measured. Since .sigma..sub.X as indicated is equal to 1/R.sub.X, the effects of the parallel and series capacitances C.sub.P and C.sub.X are inverted as shown.
Thus a circuit for adequately measuring conductivities of solutions over a wide range of conductivity values must adequately take into account and eliminate both C.sub.P and C.sub.X as sources of inaccuracy.
An early method used in the prior art to measure solution conductance employs an AC conductance bridge, where various reactance are added to arms of the bridge to compensate C.sub.X, C.sub.P or both. This method has been shown to be effective but is generally slow and requires manual operation.
Digital impedance meters are available (Model 254, Electro Scientific Industries, Inc.) that employ sine-wave excitation and synchronous phase angle detection to separate the conductance due to reactive components. Again, this method is effective but is expensive and relatively slow, and cannot be effectively automated, as would be desired.
It is desirable to use a square-wave excitation signal to drive the cell, that is, to apply a square-wave signal to one electrode of the sample cell, and measure the current through the cell to determine the resistivity of the solution, due to the ease with which precision amplitude square-waves can be generated. However, the presence of C.sub.P and C.sub.X can lead to significant linearity errors if not actively compensated for. Applicant's prior U.S. Pat. No. 4,683,435 addresses in detail one approach to compensating some of these errors while using a square-wave drive signal. The present invention reflects additional understanding of the problems inherent in measuring the conductivity of a solution confined between two electrodes in a cell and presents additional and improved solutions thereto.